A027644 -id:A027644 - cp's OEIS frontend

A027643 Numerators of poly-Bernoulli numbers B_n^(k) with k=2.

1, 1, -1, -1, 7, 1, -38, -5, 11, 7, -3263, -15, 13399637, 7601, -8364, -91, 1437423473, 3617, -177451280177, -745739, 166416763419, 3317609, -17730427802974, -5981591, 51257173898346323, 5436374093, -107154672791057, -213827575

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..521
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Index entries for sequences related to Bernoulli numbers.

Cf. A027641, A027642, A027644, A027645, A027646, A027647, A027648, A027649, A027650, A027651.

A027643:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027643(n,2): n in [0..30]]; // G. C. Greubel, Aug 02 2022

a := n -> numer((-1)^n*add( (-1)^m*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
seq(a(n), n=0..27);

k=2; Table[Numerator[(-1)^n Sum[(-1)^m m! StirlingS2[n, m]/(m+1)^k, {m, 0, n}]], {n, 0, 27}] (* Michael De Vlieger, Oct 28 2015 *)

def A027643(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027643(n,2) for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = numerator of Sum_{k=0..n} W(n,k)*h(k+1) with W(n,k) = (-1)^(n-k)*k!* Stirling2(n+1,k+1) the Worpitzky numbers and h(n) = Sum_{k=1..n} 1/k^2 the generalized harmonic numbers of order 2. - Peter Luschny, Sep 28 2017

A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

Original entry on oeis.org

1, 8, 46, 230, 1066, 4718, 20266, 85310, 354106, 1455278, 5938186, 24104990, 97478746, 393095438, 1581931306, 6356390270, 25511588986, 102304505198, 409992599626, 1642294397150, 6576150108826, 26325519044558, 105364834103146, 421647614381630, 1687155299822266

Offset: 0

N. J. A. Sloane

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{3,n}. - Vincent Pilaud, Sep 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.
Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027651.
First differences of A016269.
Row 3 of array A099594.

[6*4^n-6*3^n+2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
seq(a(n, -3), n = 0..30);

Table[6*4^n-6*3^n+2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)

Vec((1-x)/((1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015

[2^n -6*3^n +6*4^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = 6*4^n - 6*3^n + 2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1-x)/((1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: exp(2*x) - 6*exp(3*x) + 6*exp(4*x). - G. C. Greubel, Aug 02 2022

A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

Original entry on oeis.org

1, 16, 146, 1066, 6902, 41506, 237686, 1315666, 7107302, 37712866, 197451926, 1023358066, 5262831302, 26903268226, 136887643766, 693968021266, 3508093140902, 17693879415586, 89084256837206, 447884338361266, 2249284754708102, 11285908565322946, 56587579617416246

Offset: 0

N. J. A. Sloane

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{4,n}. - Vincent Pilaud, Sep 16 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.
Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Hiroyuki Komaki, Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index, arXiv:1503.04933 [math.NT], 2015.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120).

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027650.

Row n=4 of array A099594.

[24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n,j)/(j+1)^k, j=0..n);
seq(a(n, -4), n=0..30);

Table[24*5^n -36*4^n +14*3^n -2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{14,-71,154,-120},{1,16,146,1066},30] (* Harvey P. Dale, Nov 20 2019 *)

Vec((1+4*x)*((1-x)^2)/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015

[24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = 24*5^n -36*4^n +14*3^n -2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1+4*x)*(1-x)^2/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: 24*exp(5*x) - 36*exp(4*x) + 14*exp(3*x) - exp(2*x). - G. C. Greubel, Feb 07 2018

A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

Original entry on oeis.org

1, 16, 1296, 3456, 3240000, 144000, 1555848000, 59270400, 5000940000, 1587600, 9762501672000, 11269843200, 221794053611130000, 39390663312000, 5849513501832000, 519437318400, 407131014322092060000, 1063331477208000

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..807
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
M. Kaneko, Poly-Bernoulli numbers
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Index entries for sequences related to Bernoulli numbers.

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.

A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027648(n,4): n in [0..20]]; // G. C. Greubel, Aug 02 2022

a:= (n, k) -> denom( (-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m = 0..n) );
seq(a(n, 4), n = 0..30);

With[{k = 4}, Table[Denominator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* Michael De Vlieger, Mar 18 2017 *)

def A027648(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027648(n,4) for n in (0..20)] # G. C. Greubel, Aug 02 2022

a(n) = denominator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 4.

A099596 Primes p such that the denominator of the poly-Bernoulli number B(2,n) equals 8p.

Original entry on oeis.org

3, 5, 11, 17, 23, 47, 59, 83, 107, 137, 167, 179, 227, 239, 257, 263, 317, 347, 359, 383, 431, 443, 467, 479, 503, 557, 563, 587, 647, 659, 719, 797, 827, 839, 857, 863, 887, 983, 1019, 1091, 1097, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1487, 1499

Offset: 1

Ralf Stephan, Oct 27 2004

nonn

p such that A027644(p) = 8p.

f[n_] := Denominator[(-1)^n*Sum[(-1)^m*m!*StirlingS2[n, m]/(m + 1)^2, {m, 0, n}]]; l = {}; Do[p = Prime[n]; If[f[p] == 8p, AppendTo[l, p]], {n, 240}]; l (* Robert G. Wilson v, Oct 28 2004 *)

More terms from Robert G. Wilson v, Oct 28 2004

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A027643 Numerators of poly-Bernoulli numbers B_n^(k) with k=2.

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A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

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A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

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A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

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A099596 Primes p such that the denominator of the poly-Bernoulli number B(2,n) equals 8p.

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